Abstract
In a seminal paper published in 1982, Fredkin and Toffoli have introduced conservative logic, a mathematical model that allows one to describe computations which reflect some properties of microdynamical laws of Physics, such as reversibility and conservation of the internal energy of the physical system used to perform the computations. In particular, conservativeness is defined as a mathematical property whose goal is to model the conservation of the energy associated to the data which are manipulated during the computation of a logic gate.
Extending such notion to generic gates whose input and output lines may assume a finite number d of truth values, we define conservative computations and we show that they naturally induce a new NP–complete decision problem and an associated NP–hard optimization problem. Moreover, we briefly describe the results of five computer experiments performed to study the behavior of some polynomial time heuristics which give approximate solutions to such optimization problem.
Since the computational primitive underlying conservative logic is the Fredkin gate, we advocate the study of the computational power of Fredkin circuits, that is circuits composed by Fredkin gates. Accordingly, we give some first basic results about the classes of Boolean functions which can be computed through polynomial–size constant–depth Fredkin circuits.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation. Combinatorial Optimization Problems and Their Approximability Properties. Springer, Heidelberg (1999)
Beigel, R., Tarui, J.: On ACC. In: Proceedings of the 32nd IEEE Symposium on Foundations of Computer Science, pp. 783–792 (1991)
Bennett, C.H.: Logical reversibility of computation. IBM Journal of Research and Development 17, 525–532 (1973)
Cattaneo, G., Della Vedova, G., Leporati, A., Leporini, R.: Towards a Theory of Conservative Computing. Submitted for publication (2002), e-print available at: http://arxiv.org/quant-ph/abs/0211085
Cattaneo, G., Leporati, A., Leporini, R.: Fredkin Gates for Finite-valued Reversible and Conservative Logics. Journal of Physics A: Mathematical and General 35, 9755–9785 (2002)
Fredkin, E., Toffoli, T.: Conservative Logic. International Journal of Theoretical Physics 21(3/4), 219–253 (1982)
Garey, M.R., Johnson, D.S.: Computers and Intractability. A Guide to the Theory on NP-Completeness. W. H. Freeman and Company, New York (1979)
Goldmann, M., Håstad, J., Razborov, A.: Majority Gates vs. general weighted Threshold Gates. Computational Complexity 2(4), 277–300 (1992)
Hajnal, A., Maass, W., Pudlák, P., Szegedy, M., Turán, G.: Threshold Circuits of Bounded Depth. Journal of Computer and System Sciences 46(2), 129–154 (1993)
Håstad, J., Goldmann, M.: On the power of small-depth threshold circuits. Computational Complexity I 2, 113–129 (1991)
Landauer, R.: Irreversibility and heat generation in the computing process. IBM Journal of Research and Development 3, 183–191 (1961)
Leporati, A., Della Vedova, G., Mauri, G.: An Experimental Study of some Heuristics for Min Storage (2003) (submitted for publication)
Muroga, S.: Threshold Logic and its Applications. Wiley-Interscience, Hoboken (1971)
Petri, C.A.: Griindsatzliches zur Beschreibung diskreter Prozesse. In: Proceedings of the 3rd Colloquium über Automatentheorie (Hannover, 1965) pp. 121-140. Birkhäuser Verlag, Basel (1967); English translation: Fundamentals of the Representation of Discrete Processes, ISF Report 82.04 (1982)
Razborov, A., Widgerson, A.: nΩ(log n) lower bounds on the size of depth-3 threshold circuits with AND gates at the bottom. Information Processing Letters 45(6), 303–307 (1993)
Rescher, N.: Many-valued logics. McGraw-Hill, New York (1969)
Rosser, J.B., Turquette, A.R.: Many-valued logics. North Holland, Amsterdam (1952)
Roychowdhury, V.P., Siu, K.Y., Orlitsky, A.: Theoretical Advances in Neural Computation and Learning. Kluwer Academic, Dordrecht (1994)
Siu, K.Y., Bruck, J.: On the Power of Threshold Circuits with Small Weights. SIAM Journal on Discrete Mathematics 4(3), 423–435 (1991)
Siu, K.Y., Roychowdhury, V.P.: On Optimal Depth Threshold Circuits for Multiplication and Related Problems. SIAM Journal on Discrete Mathematics 7(2), 284–292 (1994)
Vollmer, H.: Introduction to Circuit Complexity: A Uniform Approach. Springer, Heidelberg (1999)
Yao, A.C.: On ACC and Threshold Circuits. In: Proceedings of the 31st IEEE Symposium on Foundations of Computer Science, pp. 619–628 (1990)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Mauri, G., Leporati, A. (2003). On the Computational Complexity of Conservative Computing. In: Rovan, B., Vojtáš, P. (eds) Mathematical Foundations of Computer Science 2003. MFCS 2003. Lecture Notes in Computer Science, vol 2747. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45138-9_5
Download citation
DOI: https://doi.org/10.1007/978-3-540-45138-9_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40671-6
Online ISBN: 978-3-540-45138-9
eBook Packages: Springer Book Archive